체 (수학)

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정의

A field is a set $F$ together with two binary operations on $F$ called addition and muliplication. A binary operation on $F$ is a mapping $F \times F \rightarrow F$.

• The addition of $a$ and $b$ in $F$ is called the sum of $a$ and $b$, and is denoted as $a + b$.
• The multiplication of $a$ and $b$ in $F$ is called the product of $a$ and $b$, and is demoted as $a \times b$.

The following properties are required to be satisfied:

• Associativity of addition and multiplication: $a + (b + c) = (a + b) + c$, and $a \times (b \times c) = (a \times b) \times c$.
• Commutativity of addition and multiplication: $a + b = b + a$, and $a \times b = b \times a$
• Distributivity of multiplication over addition: $a \times (b + c) = (a \times b) + (a \times c)$
• Additive identity and multiplicative identity: there exist two distinct elements 0 and 1 in $F$ such that $a + 0 = a$ and $a \times 1 = a$.
• Additive inverse: for every $a$ in $F$, there exists an element in $F$, denoted by $-a$, called the additive inverse of $a$, such that $a - a = 0$
• Multiplicative inverse: for every $a \neq 0$ in $F$, there exists an element in $F$, denoted by $a^{-1}$ or $1/a$, called the multiplicative inverse of $a$, such that $a \cdot a^{-1} = 1$.